1Department of Management, College of Information System and Management, National University of Defense Technology, Changsha, Hunan 410073, China2School of Software, Shenzhen Institute of Information Technology, Shenzhen 518029, China

Abstract

This paper addresses multiobjective flexible job-shop scheduling problem (FJSP) with three simultaneously considered objectives: minimizing makespan, minimizing total workload, and minimizing maximal workload. A hybrid multiobjective evolutionary approach (H-MOEA) is developed to solve the problem. According to the characteristic of FJSP, a modified crowding distance measure is introduced to maintain the diversity of individuals. In the proposed H-MOEA, well-designed chromosome representation and genetic operators are developed for FJSP. Moreover, a local search procedure based on critical path theory is incorporated in H-MOEA to improve the convergence ability of the algorithm. Experiment results on several well-known benchmark instances demonstrate the efficiency and stability of the proposed algorithm. The comparison with other recently published approaches validates that H-MOEA can obtain Pareto-optimal solutions with better quality and/or diversity.

1. Introduction

As its practical importance, job-shop scheduling problem (JSP) has received considerable attention in various application areas. JSP is one of the hardest combinatorial optimization problems and has been proven to be a NP-hard problem [1, 2]. Flexible job-shop scheduling problem (FJSP) is a generalization of the classic JSP, which allows one operation to be performed on more than one machine. Consequently, in addition to the operations scheduling, another issue that needs to be taken into account is the assignment problem, that is determining the assignment of operations to machines. Brucker and Schlie [3] were among the first to address this problem. According to the flexibility, FJSP can be generally categorized into total flexibility FJSP (T-FJSP) and partial flexibility FJSP (P-FJSP) [4, 5].

Due to the complexity of FJSP and its NP-hard characteristic, heuristic or metaheuristic methods are preferable to solve the problem. Integrated approaches have been often used in order to improve the performance of algorithms [6]. With simultaneous consideration of multiple objectives, in particular makespan, total workload, and maximal workload, the approaches can be generally categorized into three classes [7]: Transform the multiobjective problem to a monoobjective problem by a weighted sum approach; The non-Pareto approaches dealing with different objectives in a separated way; The Pareto approaches based on the Pareto optimality concept.

In the first type of approaches mentioned above, objective function is linearly combined by weighed sum approach and is given as follows:
where , , and , respectively, denote the weight coefficients for the three objective values, satisfying . According to existing literature, most approaches can be categorized as this type. Xia and Wu [6] proposed a hybrid approach by combining particle swarm optimization algorithm with simulated annealing. Zhang et al. [8] also introduced a hybridized particle swarm optimization approach for multiobjective FJSP. Liu et al. [9] presented a multiswarm approach to multiobjective FJSP. Gao et al. [10, 11] proposed two hybrid genetic algorithms. Experiment results show that Gao's algorithms are efficient for multiobjective FJSP. Xing et al. [12] designed a simulation model to deal with multiobjective FJSP, and an efficient search method [13] was developed by the authors. Li et al. [14] proposed a hybrid tabu search algorithm integrating variable neighborhood search to solve multiobjective FJSP. Bagheri et al. [15] proposed an artificial immune algorithm in which a function of makespan was taken as the affinity evaluation. Vlcot and Billaut [16] proposed a tabu search algorithm for finding a set of nondominated solution, with a linear combination of objectives makespan and maximal lateness. Kacem et al. [4] proposed a controlled evolutionary approach for multiobjective FJSP. In the proposed algorithm, objectives of makespan and total workload were dealt with separately. In particular, makespan was taken as evaluation function when the generation index is even, otherwise, total workload was taken as evaluation function.

Compared with the approaches of transforming the problem to a single-objective one, the literature on Pareto-based approaches, especially the multiobjective evolutionary approaches, rather scant. Kacem et al. [5] proposed a Pareto approach based on the hybridization of fuzzy logic and evolutionary algorithms to solve the FJSP. Ho and Tay [17] presented an efficient approach combining evolutionary algorithm and guided local search. Frutos et al. [18] introduced a Memetic Algorithm based on the nondominated sorting genetic algorithm (NSGA-II) [19], in which a local search procedure (simulated annealing) was added. Wang et al. [20] solved FJSP by a multiobjective genetic algorithm based on immune and entropy principle. Li et al. [21] introduced a hybrid Pareto-based discrete artificial bee colony algorithm for solving multiobjective FJSP, and, in their subsequent research, Li et al. [22] presented a hybrid Pareto-based local search (PLS) embedding a variable neighborhood search-based self-adaptive strategy for the problem. Moslehi and Mahnam [23] proposed a Pareto approach hybridizing particle swarm and local search to solve multiobjective FJSP.

In this paper, we propose a hybrid multiobjective evolutionary algorithm (H-MOEA) to solve the FJSP. In the proposed H-MOEA, NSGA-II [19] is employed as the basic optimizer. In multiobjective optimization problems, convergency and diversity are two important issues. The former indicates the algorithm's ability to find the true Pareto optimal solutions, and the latter reflects the algorithm's ability to find as much as possible different Pareto optimal solutions. However, in FJSP, there are not many Pareto optimal solutions in objective space. Thus, a modified crowding distance measure is developed to keep the diversity of the population. In order to improve the convergency of the algorithm, a local search based on critical path theory is incorporated into the H-MOEA. Additionally, well-designed chromosome representation and genetic operators are presented in the algorithm.

The remainder of this paper is organized as follows. In Section 2, we briefly introduce the problem formulation and optimization model of FJSP. Some preliminary concepts are presented in Section 3. The elements of the proposed H-MOEA are described in Section 4. The computational results and the comparison with other algorithms are reported in Section 5. Finally, some conclusion of our work is presented in the last section.

2. Problem Formulation

Similar to the classic JSP, FJSP takes into account the assignment of each operation to a machine and sets its starting and ending times. However, the task is more challenging than the classic one because it requires a proper selection of a machine from a set of machines to process each operation. A FJSP considering jobs to be processed on machines can be described as follows.(1)There is a set of jobs to be scheduled. (2)The set of machines is denoted as , where is the number of machines. (3)Each job consists of a sequence of operations .(4)The execution of operation requires one machine out of a set of given machines represented as .

FJSP is to determine an assignment and a sequence of operations on machines to minimize(1)makespan, which indicates the maximal completion time of all jobs, denoted as ; (2)total workload, which represents the total working time of all machines, denoted as ; (3)critical machine workload, which is the most workload among all machines, denoted as ;

Then, the multiobjective FJSP can be generally formulated as follows:

There are some assumptions and constraints in FJSP.(1)Jobs are independent from each other. (2)Machines are independent from each other. (3)Setting up time of machines is negligible. (4)Move time between operations is negligible. (5)At a given time, a machine can execute at most one operation. (6)No more than one operation of the same job can be executed at a time. (7)There are no precedence constraints among the operations of different jobs.

3. Preliminary Concepts

3.1. Concept of Domination and Nondomination

Within the framework of multiobjective optimization, the concept of dominance is defined as follows: a solution dominates solution if is as good as on all objective measures and better than on at least one objective measure. Thus, in this paper, we formally define the schedule dominance as follows.

Definition 3.1. A schedule dominates schedule if , with strict inequality holding for at least one objective measure.

Thus, the nondominance concept can be defined as follows.

Definition 3.2. A schedule is said to be a nondominated schedule if it is not dominated by any other individual in the population.

3.2. Concept of Critical Path

Feasible schedules of FJSP can be represented as a directed graph , where is the operation set, represented as node set in the graph, is the conjunctive arc set, and is the disjunctive arc set. Thus, some graph theory is often integrated into the optimization algorithm of FJSP. In particular, local search procedure based on critical path theory is usually combined with heuristic or meta-heuristic.

Two important concepts in directed solution graph are earliest start time and latest start time of a node [11]. The first concept is used to indicate the earliest time at which an operation can be processed, while the second concept is related to the latest time at which the implementation of an operation can begin without delaying the makespan of a schedule. For an operation, if its earliest start time is equal to latest start time, any delay in the start of the operation will delay the completion of the schedule. Such an operation is critical to the completion of all the jobs on time [11]. A path from starting node to ending node in a graph that entirely consists of critical operations is called a critical path. The makespan of a schedule is equal to the length of the critical path [24].

3.3. Concept of Critical Block

A critical block is a set of consecutive operations on the critical path processed by the same machine [17]. One machine can be associated with more than one critical block on a given critical path. The first operation of the critical block is called as block head, while the last operation is termed as block rear. The operation between block head and rear is called as internal block. Figure 1 shows an example of critical block.

Figure 1: An example of critical block.

4. Methodology

It is clear that a multiobjective approach is needed to solve the problem. To perform this task, one of the classic MOEAs, named NSGA-II [19], is employed as the multiobjective optimization algorithm. In the procedure of NSGA-II, the parent population and offsprings are combined and sorted in order to generate a population for the next generation. A nondominated sorting mechanism is performed to classify the combined population into different ranks of nondomination. A crowding-distance assignment is employed to ensure diversity is maintained among nondominated solutions. However, the crowding-distance mechanism is not very effective when there are few nondominated solutions and many-to-one mapping from decision space and objective space exists. Thus, a modified crowding-distance measure is introduced to maintain the diversity of the population. Additionally, chromosome representation, crossover, and mutation operators in original NSGA-II are redesigned for FJSP in this research.

4.1. Modified Crowding-Distance Measure

In multiobjective optimization problem, along with convergence to the Paretooptimal set, another issue that needs to be addressed is a good spread of solutions in the obtained set of solutions. To do this, the population should have a good diversity preservation. The original NSGA-II [19] employs a crowded-comparison approach to preserve the diversity of the population. The crowding distance of a solution is measured as the average distance of the solution from its nearest neighbors with the same nondomination rank in the objective space. Shown as in Figure 2, the crowding distance is calculated as the average side length of the cuboid.

Figure 2: Crowding distance calculation.

However, in FJSP, there are not many nondominated solutions [17]. Furthermore, there exists many-to-one mapping from decision space to objective space. For example, two solutions with different machine assignment and/or operation sequence may have the same measure on objectives (an example will be shown in Section 5, Figures 7 and 8). In such a case, the crowding distance measure in original NSGA-II cannot effectively indicate the diversity estimation of an individual in the population. Shown as in Figure 3, suppose solutions A, B, and C in decision space have same machine assignments but different operation sequences, while solution D has different machine assignment and operation sequence from the other three solutions. In objective space, the four solutions have the same measures on all objectives. Based on the definition of nondomination solution, all of the four solutions will be maintained in the population. According to the crowding distance measure in the original NSGA-II, the four solutions have the same value on crowding distance. It indicates that the four solutions have the same priority to be selected to form the next population. However, it is reasonable to consider that solution D has a better contribution to the diversity of population. Thus, solution D should have a higher priority in the selection operator and the operator to form the population of next generation.

Figure 3: An example of many-to-one mapping from decision space to objective space.

We modify the crowding distance as a measure in decision space. Such an idea has received much attention in recent years and been employed in some literatures related to multiobjective evolutionary algorithms [25–27]. Due to the stochastic feature of the nature-inspired algorithms, in FJSP, different solutions might have the same machine assignment. In order to keep the diversity of machine assignments, the crowding distance is measured by a function of the times of an assignment that occurs in the population. It is simply calculated as follows:
where is the size of population, represents the number of individuals in the population which have the same machine assignment with individual . According to the modified crowding distance measure, the solution with a diverse machine assignment will be selected first in the nondomination front to form the next population.

4.2. Chromosome Representation

Since FJSP has two independent tasks: machine assignment and operation scheduling, the chromosome representation of a solution should contain the information of these two parts. Permutation representation is often used to express operation sequence. In order to tackle the precedence constraints, Gen et al. [28] name all operations for a job with the same symbol and then interpret them according to the order of occurrences in the sequence of a given chromosome. This representation has been employed in FJSP, and a two-vector representation was proposed by Gao et al. [10, 11]. This paper also represents the chromosome in a two-vector approach, named machine assignment vector and operation sequence vector, but slightly modifies the representation of operation sequence.

In this study, the operation sequence is represented in a permutation way. Each operation is identified by a unique operation ID. A schedule is generated with the consideration of precedence constraints. Since we assumed that there are no precedence constraints among the operations of different jobs, each operation at most has one predecessor and successor, respectively. Figure 4 shows the modified two-vector chromosome representation of a FJSP with 4 jobs, 12 operations, and 4 machines. The length of each chromosome is equal to the total number of operations, denoted as . For each position in the chromosome, the machine assignment vector represents the machine selected for the operation on position , the operation sequence vector denotes the ID index of the operation.

Figure 4: An example of chromosome representation.

4.3. Population Initialization

The algorithm starts with initialization of population. For each individual, both machine assignment and operation sequence need to be initialized.

4.3.1. Machine Assignment Initialization

Machine assignment is initialized by a mix of different assignment rules. Here, we employ two assignment rules proposed in Kacem et al. [4].(I)Assignment Rule 1: search for the global minimum in the processing time table. (II)Assignment Rule 2: randomly permute jobs and machines in the processing time table.

Assignment Rule 1 starts from the operation that corresponds to the global minimum in the processing time table [29]. After several iterations, all assignments are found by entirely exploring the table of processing times [4]. Assignment Rule 2 randomly permutes the jobs and the machines before applying the approach by localization [29]. Thus, Assignment Rule 2 can avoid the inconvenience that tends to privilege some configurations as compared to the others [4]. The initial set of assignments is generated by making use of a mix of these two assignment rules. In the experiments of this research, we employ the same assignment initialization proportion as in [29], that is, 10% of initial population will be generated by Assignment Rule 1 and 90% of it will be generated by Assignment Rule 2. For details about these two rules, readers are referred to Kacem et al. [4] and Pezzella et al. [29].

4.3.2. Operation Sequence Initialization

Once the assignments are settled up, the sequence of operations on the machine needs to be determined. In the proposed H-MOEA, mix of the following four rules is used to select an operation from an eligible operation set. (i)Long Processing Time (LPT), process the operation with maximal processing time firstly. (ii)Most Work Remaining (MWR), process the operation among the task with the maximal total processing time firstly. (iii)Most Operation Remaining (MOR), process the operation among the task with most remaining operation firstly. (iv)Randomly Selection (RS), randomly select an operation to be scheduled from the eligible set.

4.4. Decoding

A shift is called global left-shift if some operations can be started earlier in time without delaying any other operations even though the shift has changed the operation sequence. A schedule is said to be active if no global left-shift exists [10]. Since makespan is taken as one of the optimization objective, in this study, we construct a schedule as active one, that is, the schedule is constructed as compact as possible.

In order to transfer the chromosome into an active schedule, several scheduling techniques are combined to generate the schedule. In concrete, we integrate Serial Sequence Generation Schedule (SSGS) [30, 31], scheduling algorithm proposed by Kacem et al. [4] and priority-based decoding method introduced by Gao et al. [11] in the decoding procedure. In the proposed decoding method, operations are scheduled at their earliest time from left to right according to their ID index in the chromosome. Algorithm 1 shows the proposed decoding procedure.

Algorithm 1: Pseudocode of decoding procedure.

4.5. Crossover and Mutation

Crossover and mutation operators are used to generate the offspring, the former operator applies to pairs of chromosomes, while the latter operator applies to single one. Since the chromosome of FJSP consists of two vectors, genetic operators are executed to each part independently.

4.5.1. Assignment Operators

Assignment operators change the machine assignments of individuals, while the operation sequences are preserved.

We adopt crossover presented in Kacem et al. [4], shown as in Algorithm 2. is the job number, and and are the operation numbers of job and , respectively.

Algorithm 2: Procedure of crossover for assignment.

It would be interesting to make genetic operators able to contribute in optimization objectives [4]. Along this avenue, we propose several mutation operators for machine assignment.(1)Mutation operator balancing workload of machines, denoted as Assignment_MO1. This mutation operator is adopted from Kacem et al. [4] but made a slight modification. The operation on most loaded machine is randomly selected and assigned to another machine such that the workload of the new assigned machine cannot exceed the maximal workload of the individual before applying mutation operation. (2)Mutation operator reducing total workload, denoted as Assignment_MO2. This mutation operator aims at contributing to optimization of total workload. A certain number of operations are randomly selected to be assigned on another machine (if exists), satisfying the following two conditions: (a)the processing time of the operation on the new assigned machine is less than the original machine;(b)the workload of the new assigned machine cannot exceed the maximal workload of the individual. (3)Mutation operator reassigning critical operations, denoted as Assignment_MO3. In this operator, a certain number of critical operations are randomly selected and assigned to another machine on which the processing time cannot exceed the processing time on the original machine. In other words, the total workload after mutation is equal to or less than original one. (4)Immigration mutation operator, denoted as Assignment_MO4. This operator is adopted from Gao et al. [11], which randomly generates new machine assignment for individuals according to the rules discussed in the stage of assignment initialization.

4.5.2. Sequence Operators

In sequence operators, operation sequences are changed, while the machine assignments are preserved. Several crossover operators have been reported in the literature for permutation representation. In our algorithm, we apply a two-point position-based crossover [32, 33] for operation sequence. Usually, the two-point position-based crossover for scheduling problem can be divided into two versions [32]: in the first version, the activities outside the two selected points are inherited from one parent to the child, while in the other version, the set of activities between the two randomly selected points is inherited from one parent to the child. In this research, we employ the first version of crossover operator. In other words, for each crossover, two crossing points will be generated. We randomly generate these two crossing points from the interval , denoted as and , . Then, parent chromosome is separated as three parts by crossing points. For instance, we assume that two parent chromosomes and are selected for crossover, which can be represented as and , respectively. Two children and are constructed by interchangeably inheriting different parts from parents, that is, takes the first and third parts from parent , while inherits the second part from . The process is the same for generating , but with a reverse sequence . What should be carefully noted is that, in order to preserve the precedence constraints, when inheriting the second and third parts, all operations that are already in the previous parts should be eliminated. This crossover operator is widely used in other scheduling problems, for example, project scheduling problem [34]. Hartmann [35] showed that this operator creates precedence feasible operation sequences.

Mutation operator generates new individuals slightly different from the parent. In this study, we adopt the procedure presented by Shadrokh and Kianfar [36] as the basic idea of mutation operator for operation sequence. In this mutation operator, an element in the operation sequence is randomly selected and inserted into another position between the interval determined by its last predecessor and first successor. Since this mutation operator is introduced for project scheduling with a permutation representation, it might not work effectively for FJSP in some cases. For example, when the machine predecessor and successor of the selected operation are not between its job predecessor and successor, changing the position of the selected operation will not impact the performance of an individual. Thus, we propose two mutation operators for FJSP based on the permutation representation.

The first mutation operator, denoted as Sequence_MO1, is used to permutate the position of an operation's job successor and/or job predecessor if its machine successor and/or machine predecessor are out of the interval of job predecessor and successor. Figure 5 shows an example of mutation operator Sequence_MO1. Suppose operation is the selected operation to be applied mutation operator. The job successor and machine successor of are and , with a index of 7 and 8, respectively, in the operation sequence. In such a case, permutating the position of in the sequence between its job predecessor and successor cannot change the schedule sequence of its machine successor . Then, mutation operator Sequence_MO1 is to permutate the position of the job successor to a feasible interval after the position of machine successor (if exists). The situation of predecessor is the same with successor.

Figure 5: The first mutation operator for operation sequence.

The second mutation operator for operation sequence, denoted as , is defined as permutating the position of the first critical operation in a randomly selected interval from the operation sequence. This operator bears much similarity with the one proposed in Shadrokh and Kianfar [36]; the only difference is that is only applied to critical operation.

4.6. Local Search Approach

The evolution speed of simple GAs is relatively slow [37]. One of the possible promising avenues to improve the convergence speed is the hybridization of local search [11]. Ishibuchi and Murata [38, 39] were among the first to implement such a hybridization. A central problem of any local search procedure for combinatorial optimization problem is how to define the effective neighborhood around an initial solution [10]. Since FJSP consists of two subproblems, that is, machine assignment and operation sequence, the neighborhood of a solution of FJSP should be defined to indicate the neighbors of these two parts.

4.6.1. Assignment Neighborhood

The makespan of a schedule is invariant when moving operations that are not on critical paths to other machines [17]. Thus, an efficient neighborhood of machine assignment is often constructed by adjusting critical operations on a machine to other available machines, such as in Li et al. [22]. In this study, an efficient assignment neighborhood is presented based on the idea of critical operation adjustment.

Step 1. Randomly select a machine on which the workload is equal to the maximal workload of the individual.

Step 2. Select the critical operation(s) on machine to be adjusted according to given probabilities based on the number of operations belonging to the same job. Suppose the vector of critical operations on machine is denoted as , the numbers of operations belonging to the same job with a critical operation on machine are represented as . Then, the probability vector of adjusting each critical operation is given as , where is a coefficient, which is set as , is the average number of operations for each job.

Step 3. For each critical operation on machine selected to be executed for machine adjustment, assign the operation to a new machine such that the combined value of total workload and maximal workload after adjustment is minimum. Let and , respectively, denote the total workload and maximal workload of the individual after machine assignment. For a selected critical operation, search for a machine which is capable of processing the operation such that the combined function is minimum among all possible machines.

4.6.2. Sequence Neighborhood

In order to define the neighborhood of operation sequence, critical path and/or critical block are often employed in FJSP by many researchers [8, 10, 11, 14, 17, 21, 22, 40]. Along this avenue, in this study, the effective neighborhood of operation sequence is based on the concept of critical block. A neighbor solution is obtained by adjusting the position of a randomly selected operation which belongs to a critical block. Algorithm 3 shows the procedure of sequence adjustment.

Algorithm 3: Procedure of sequence adjustment.

4.6.3. Framework of Local Search

Xing et al. [13] proposed an efficient search method for FJPS. In that proposed approach, two feasible moves were introduced to machine assignment, followed by a procedure of operation sequence. Then, a best move with a optimal performance can be obtained after feasible moves evaluation. Based on this approach, the present research proposes a local search procedure which incorporates machine adjustment and sequence adjustment. Algorithm 4 shows the procedure of the proposed local search. The local search procedure is implemented on each individual in the population. The size of neighborhood is set as dependent with the value of nondominated rank of an individual, given as follows:
where JobNum represents the number of jobs and ranki indicates the nondominated rank value of the ith individual in the population, Round() is the function to obtain the integer part of a number. However, with the consideration of computational cost, this local search procedure is only implemented every dth generation. Several important issues in the hybridization of MOEAs and local search, such as balance between genetic search and local search [41, 42] and choice of solutions to which local search is applied [43], are left in our future work to study more efficient algorithms to solve larger scale problems.

Algorithm 4: Procedure of local search.

4.7. Framework of H-MOEA

By hybridizing local search with multiobjective genetic algorithm, the convergence speed to local optimal can be improved. On the other hand, it might increase computational time per iteration. Thus, a well-designed algorithm is very important to the real application cases. Figure 6 illustrates the framework of the proposed H-MOEA.

5. Computational Results

In this section, we compared the results obtained by the proposed hybrid multiobjective evolutionary approach (H-MOEA) and other recently published algorithms. In particular, we compared the results of AL+CGA and FL+EA by Kacem et al. [4, 5], PSO+SA by Xia and Wu [6], the hybrid genetic algorithm (hGA) by Gao et al. [11], the multiobjective evolutionary algorithm with local search (MOEA+GLS) by Ho and Tay [17], the multiobjective genetic algorithm (MOGA) proposed by Wang et al. [20], the PSO+TS by Zhang et al. [8], the artificial immune algorithm (AIA) by Bagheri et al. [15], the Pareto-based discrete artificial bee colony algorithm (P-DABC) by Li et al. [21], the Pareto-based local search algorithm (PLS) developed by Li et al. [22], and the multiobjective particle swarm optimization and local search method proposed by Moslehi and Mahnam [23]. We also compared the results obtained by the present approach with some algorithms developed by our group, such as local search method developed by Xing et al. [12].

Moreover, in the stage of operation sequence initialization, the percentages of different rules were set as follows: Long Processing Time (LPT) 30%; Most Work Remaining (MWR) 30%; Most Operation Remaining (MOR) 30%, Randomly Selection (RS) 10%. The local search procedure is applied every 10th generation.

5.2. Test Instances I

The first five test instances, range from 4 jobs 5 machines to 15 jobs 10 machines, were taken from Kacem. Release date was not considered in this set of test instances. The three simultaneously considered objectives are, respectively, denoted as (minimization of makespan), (minimization of total workload), and (minimization of critical workload). Table 1 shows the comparison of the obtained results on the five instances. In order to investigate the performance of proposed K-MOGA, we preprint the solutions dominated by the ones obtained by K-MOEA in italic font. From Table 1, one can see that our algorithm performs better for at least one problem instance, compared to all other algorithms expect MOEA-GLS. By looking at the results obtained by MOEA-GLS and our algorithm, it is shown that our algorithm is competitive to MOEA-GLS, except in problem where 4 nondominated solutions were obtained by MOEA-GLS, while just 3 ones were obtained by our algorithm. The average processing time (in second), denoted as , with 20 runs is reported in the last column in Table 1. The values of indicate the efficiency of our algorithm.

Table 1: Comparison of the Kacem instances without release date.

As discussed earlier, it exits many-to-one mapping from decision space to objective space in FJSP. Figures 7 and 8 show two solutions of instance. These two solutions have the same objective values, makespan 11, total workload 91, and maximal workload 11, but with different machine assignments. This indicates that keeping diversity in decision space could contribute to the convergency of the algorithm towards to Paretooptimal front.

The performance analysis of the algorithms not only considers the end state, but it is quite important to understand how they behave during the time of evolution. In this paper, we propose to measure the mean ideal distance (MID) [44] over time. MID is used to evaluate the closeness of solutions in a nondominated set with an ideal point which is usually considered as (0,0,0) [45]. MID can be calculated as follows:
where NoN is the number of nondominated solutions, and represents the distance of the th nondominated solution from the ideal point.

Figure 9 depicts the performance curve of MID for instance without release dates. The X-axis indicates the evolving generation, and the Y-axis represents the average value of MID over 20 independent runs. To such an instance, H-MOEA converged quickly in the first 20 generation. The first nondominated solution included in the final population was found at 32 generation, and all of the final nondominated solutions were identified at 46 generation. This suggests that the proposed H-MOEA is very efficient in solving FJSPs.

Table 2 gives the comparison results with several recently published algorithms. It can be seen from Table 2 that our algorithm can obtain better or equiponderant solutions compared with others, except for instance . In the problem with consideration of release date, our algorithms obtained two nondominated solutions with an average time 10.21 s. The algorithm P-DABC proposed by Li et al. [21] could obtain 3 nondominated solutions. However, since the authors of P-DABC did not report the average processing time on this instance, it could not be considered that our algorithm performed worse on obtaining diverse nondominated solutions.

Table 2: Comparison of the Kacem instances with release date.

5.4. Test Instances III

In order to make a further investigation on the performance of our algorithm, we ran H-MOEA on several instances from the BR data set presented by Brandimarte [46]. We used the same four different scale instances as in Li et al. [21] to verify the performance of H-MOEA. Four recently published algorithms were compared to our algorithm. The first algorithm was developed by the researchers in our group Xing et al. [13], denoted as X-SM. The second algorithm AIA was proposed by Bagheri et al. [15]. The third one, which was also the most recently published one, was introduced by Li et al. [22] and denoted as PLS. The fourth algorithm, which was also the most recently published one, hybrid shuffled frog-leaping algorithm (HSFLA), was proposed by Li et al. [47]. The comparison results are reported in Tables 3, 4, 5, and 6. In order to show the performance of our algorithm in a more clear way, we also preprint the solutions dominated by other ones in italic font and preprint the solutions dominating at least one other solution in bold face.

Table 3: Comparison of the MK01 instance.

Table 4: Comparison of the MK02 instance.

Table 5: Comparison of the MK03 instance.

Table 6: Comparison of the MK08 instance.

As shown in Tables 3 to 6, compared with X-SM, H-MOEA could obtain dominating or nondominated solutions with less computational effort. The solutions obtained by AIA for all tested instances were dominated by the solutions by our algorithm. By looking at the Pareto-based approach PLS, one can see that most of solutions obtained by PLS were dominated by the ones obtained by our algorithm, while in reverse, just one solution obtained by our algorithm for instance MK02 was dominated by the solution obtained by PLS. Also, it can be seen that H-MOEA could obtain a richer nondominated set than PLS. Since the computational time of PLS was not reported by the authors, further comparison and more accurate conclusion about the performances of the two algorithms could not be available. The comparison between HSFLA and the proposed H-MOEA shows that most of nondominated solutions obtained by HSFLA were dominated by the solutions obtained by H-MOEA. It suggests that H-MOEA has better convergence ability than HSFLS on the test instances.

6. Conclusion

As its practical importance and complexity, multiobjective flexible job-shop scheduling problem (FJSP) has received considerable attention during last few years, and various heuristic or meta-heuristic approaches were reported in the literature. In this paper, we present a multiobjective evolutionary approach for solving the problem. In our algorithm, a permutation based chromosome representation and several efficient genetic operators are designed for FJSP. For multiobjective optimization problems, an important issue is obtaining well spread nondominated solutions. In other words, good diversity should be maintained in the population. To do this, a modified crowding distance measure is introduced in our approach to indicate the diversity of individuals in decision space. In order to speed up the convergency of the algorithm, a local search procedure is embedded in the evolutionary process. Experiment results indicate the effectiveness of the proposed approach. In order to handle more complex problems, in future work, the authors will address the design of more efficient hybrid multiobjective evolutionary algorithms, in which several mentioned issues, that is, balance between genetic search and local search and choice of solutions to which local search is applied, will be considered in the implementation of algorithms. In this research, it is assumed that all parameters and data are deterministic. However, in real applications, schedules are often confronted with uncertain factors. In other words, the executing environment of a schedule might be stochastic or dynamic. The authors will address the efficient approach for stochastic version of FJSP in future work.

Acknowledgments

This research was supported in part by the China Scholarship Council and National Natural Science Foundation of China under the Contracts no. 70971131, 70901074, 71001104, 71101150, and 71101096.

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